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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** #<style> .doc { font-family: monospace; white-space: pre; } </style># **)
(** Constants for under/over, to rewrite under binders using "context lemmas"
Note: this file does not require [ssreflect]; it is both required by
[ssrsetoid] and *exported* by [ssrunder].
This preserves the following feature: we can use [Setoid] without
requiring [ssreflect] and use [ssreflect] without requiring [Setoid].
*)
Require Import ssrclasses.
Module Type UNDER_REL.
Parameter Under_rel :
forall (A : Type) (eqA : A -> A -> Prop), A -> A -> Prop.
Parameter Under_rel_from_rel :
forall (A : Type) (eqA : A -> A -> Prop) (x y : A),
@Under_rel A eqA x y -> eqA x y.
Parameter Under_relE :
forall (A : Type) (eqA : A -> A -> Prop),
@Under_rel A eqA = eqA.
(** [Over_rel, over_rel, over_rel_done]: for "by rewrite over_rel" *)
Parameter Over_rel :
forall (A : Type) (eqA : A -> A -> Prop), A -> A -> Prop.
Parameter over_rel :
forall (A : Type) (eqA : A -> A -> Prop) (x y : A),
@Under_rel A eqA x y = @Over_rel A eqA x y.
Parameter over_rel_done :
forall (A : Type) (eqA : A -> A -> Prop) (EeqA : Reflexive eqA) (x : A),
@Over_rel A eqA x x.
(** [under_rel_done]: for Ltac-style over *)
Parameter under_rel_done :
forall (A : Type) (eqA : A -> A -> Prop) (EeqA : Reflexive eqA) (x : A),
@Under_rel A eqA x x.
Notation "''Under[' x ]" := (@Under_rel _ _ x _)
(at level 8, format "''Under[' x ]", only printing).
End UNDER_REL.
Module Export Under_rel : UNDER_REL.
Definition Under_rel (A : Type) (eqA : A -> A -> Prop) :=
eqA.
Lemma Under_rel_from_rel :
forall (A : Type) (eqA : A -> A -> Prop) (x y : A),
@Under_rel A eqA x y -> eqA x y.
Proof. now trivial. Qed.
Lemma Under_relE (A : Type) (eqA : A -> A -> Prop) :
@Under_rel A eqA = eqA.
Proof. now trivial. Qed.
Definition Over_rel := Under_rel.
Lemma over_rel :
forall (A : Type) (eqA : A -> A -> Prop) (x y : A),
@Under_rel A eqA x y = @Over_rel A eqA x y.
Proof. now trivial. Qed.
Lemma over_rel_done :
forall (A : Type) (eqA : A -> A -> Prop) (EeqA : Reflexive eqA) (x : A),
@Over_rel A eqA x x.
Proof. now unfold Over_rel. Qed.
Lemma under_rel_done :
forall (A : Type) (eqA : A -> A -> Prop) (EeqA : Reflexive eqA) (x : A),
@Under_rel A eqA x x.
Proof. now trivial. Qed.
End Under_rel.
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